These are the kinds of questions we should encourage the students to generate.
They should be shown two or more conflicting definitions in textbooks and
decide, as this discussion has, what would change if we altered the definition.
Then they could go on to answer Cynthias initial question. What a great geometry
course that would be.
Then they will really understand what math is and what mathematicians do..and
not just gifted kids either....but any students who can pass a geometry course.
Great discussion!!!

This material is based upon work supported by the
National Science Foundation under Grant DUE-0226284.
Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s)
and do not necessarily reflect the views of the
National Science Foundation.